Mathematical Croquet: Exploring Collisions and Pi

Introduction

• The lecture discusses a scenario involving two sliding blocks and a wall.
• Assumptions:
  • No friction
  • Perfectly elastic collisions (no energy loss)
• Aim: Count the number of collisions ("clacks") that occur in various scenarios.

Basic Scenario

• Same Mass Blocks:
  • First block hits second block:
    • Transfers all momentum to the second block.
  • Second block bounces off the wall:
    • Transfers momentum back to the first block.
  • Total Clacks: 3

Different Mass Ratios

• Case 1: First block 100 times the mass of the second:
  • Total collisions: 31
• Case 2: First block 10,000 times the mass of the second:
  • Total collisions: 314
• Case 3: First block 1,000,000 times the mass of the second:
  • Total collisions: 3,141

Pattern Identification

• Observed pattern: When the mass of the first block is a power of 100 times the second, the number of collisions corresponds to the digits of pi.
• Discovery Credit:
  • Originally discovered by mathematician Gregory Galperin in 1995, published in 2003.
  • Mentioned by viewer Henry Cavill.

Algorithm for Computing Pi

• Steps to compute digits of pi using this phenomenon:
  1. Implement a physics engine.
  2. Choose the number of digits (d) of pi to compute.
  3. Set mass of first block to be 100^(d-1).
  4. Count collisions.
• Example calculation for 20 digits:
  • Large block's mass = 100 billion billion billion billion times the mass of the smaller block (1 kg).
  • This results in counting 31 billion billion collisions.
  • Clack frequency: 100 billion billion billion billion clacks per second.

Implications and Observations

• Theoretical implementation yields an impractical approach to computing pi.
• Highlights the inefficiency of the method despite its elegance.
• Raises the question: Why does pi appear in this context?
  • Pi is usually associated with circular geometry, hinting at a hidden circle in the problem.
• The connection to conservation of energy will be explored in the next video.

Conclusion

• Encouragement to explore the problem collaboratively.
• Excitement for the next discussion about the underlying principles involving pi.
• Thank you for watching!